A function f(x) is defined as follows

$$f(x) = \left\{ {\matrix{ {x + \pi ,} & {for\,x \in [ - \pi ,0)} \cr {\pi \cos x,} & {for\,x \in \left[ {0,{\pi \over 2}} \right]} \cr {{{\left( {x - {\pi \over 2}} \right)}^2},} & {for\,x \in \left( {{\pi \over 2},\pi } \right]} \cr } } \right.$$

Consider the following statements

1. The function f(x) is continuous at x = 0.

2. The function f(x) is continuous at $$x = {\pi \over 2}$$

Which of the above statements is/are correct?

A function f(x) is defined as follows

$$f(x) = \left\{ {\matrix{ {x + \pi ,} & {for\,x \in [ - \pi ,0)} \cr {\pi \cos x,} & {for\,x \in \left[ {0,{\pi \over 2}} \right]} \cr {{{\left( {x - {\pi \over 2}} \right)}^2},} & {for\,x \in \left( {{\pi \over 2},\pi } \right]} \cr } } \right.$$

Consider the following statements

1. The function f(x) is differentiable at x = 0.

2. The function f(x) is differentiable at x = $${\pi \over 2}$$.

Which of the above statements is/are correct?

If $$\mathop {\lim }\limits_{x \to 0} \phi (x) = {a^2}$$, where a $$\ne$$ 0, then what is $$\mathop {\lim }\limits_{x \to 0} \phi \left( {{x \over a}} \right)$$ equal to ?

What is $$\mathop {\lim }\limits_{x \to 0} {e^{ - {1 \over {{x^2}}}}}$$ equal to ?